Abstract
Class field theory, i.e., the description of Abelian coverings of 1-dimensional schemes in terms of ideles, has two generalizations in two different directions. One is the higher-dimensional class field theory of Parshin, Kato, Bloch and Saito [P 1–2], [K], [B1], [Sa] which describes Abelian coverings of schemes of absolute dimension n in terms of Milnor K n -groups of the appropriately defined ring of adeles (in the classical case the group of ideles can be seen as K 1). The other generalization, the Langlands program, concerns only 1–dimensional schemes but describes higher-dimensional representations of the Galois groups in terms of representations of the groups of adelic matrices. One would like to have a common generalization of these two theories which would describe higher-dimensional representations of the Galois groups of higher-dimensional schemes. This question, although very natural, has never been discussed in the literature, even at the most rough and heuristic level (like what kind of structures should be involved in the “Langlands theory for higher dimensional schemes”). The present paper is an attempt to do so and to generate a rough conjectural framework for such a theory. I am aware that the conclusions are preliminary at best, but I hope that the general approach sketched here will help to formulate a more detailed program.
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Kapranov, M.M. (1995). Analogies between the Langlands Correspondence and Topological Quantum Field Theory. In: Gindikin, S., Lepowsky, J., Wilson, R.L. (eds) Functional Analysis on the Eve of the 21st Century. Progress in Mathematics, vol 131/132. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2582-9_4
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